Seminar in Geophysical Inverse Theory

This course is designed for graduate students and upper-level undergraduate majors (with instructor permission)

This course serves as an introduction to Geophysical Inverse Theory, built up from first principles. It contains discussion of statistics, error, and strategies to frame and solve inverse problems.

Topics to be covered:

  • Inverse and forward problems

  • Model parameterisation

  • Probability and measurement error

  • Minimum error solutions: L2 norm and simple least squares, weighted least squares

  • Minimum length solutions

  • Prior information and Bayes theorem

  • Resolution (of model and data) and generalised inverses

  • Principle of maximum likelihood

  • Non-uniqueness and localised averages

  • Factor analysis (including the VariMax principle) and Empirical Orthogonal Functions

  • Vector spaces and singular value decomposition

  • Equality and inequality constraints in inverse problems

  • Nonlinear problems – grid searches, Monte Carlo, Newton’s method, simulated annealing and Markov chains

  • Hierarchical and transdimensional problems

Instructor: Zach Eilon
Email: eilon@ucsb.edu
Office: Webb 2116

Prerequisites:   
Linear Algebra; some MATLAB(/similar) experience would be very helpful

Textbook:     
None required, but will heavily use Geophysical Data Analysis: Discrete Inverse Theory, 3rd Ed, 2012, by William Menke

Work originality:
I encourage you to discuss problems for this class with your classmates, but it is absolutely imperative that any work you submit for this class is your own. Plagiarism, defined as an attempt by a student to represent the work of another as her or his own is strictly against the policies of academic fairness and integrity of this University. This means you must very clearly attribute any quotations or copied figures (citing name + year + publication of any sources). You should always mention any classmates with whom you have collaborated (a brief marginal note will suffice), and it is not EVER permitted to copy another student’s work. If you are found to be in violation of this policy, there are very serious consequences.

Guide to problem sets:
I expect the work you turn in to be of high and careful quality. This means it should be legible and neatly presented, with effort made commensurate with the assigned task. You should always be aiming to do the assignments full justice, rather than trying to get away with the minimum required effort. Make sure your answers are complete, and that you have addressed all questions asked. Some more specific pointers:

  • Always show your full working for mathematical problems. As well as making it much easier to judge where/if you made any errors, I will not award full marks if the logic and work-flow of the answer is not clear.

  • Make sure to properly highlight your final answer to each problem

  • Answers should be mathematically correct. I.e. if you write an “equals sign”, both things on either side of it must be equal! This sounds obvious, but is often not done, leading to avoidable errors. Get in the practice of being punctilious with your mathematics :)

  • Check your answers by doing quick, back-of-the-envelope calculations to determine whether the answer you have given is in the right order of magnitude. Always make sure it passes the “smell test”; does the answer seem like it could be right? E.g., if you are supposed to be calculating crustal thickness, and your answer is 2 m or 200 km, it is surely wrong!